As is known in the art, a dynamical system is a system having output values which vary with respect to time. The time-changing output values at one time are interrelated with those at other times.
A linear dynamical system is a system in which a relatively small change in an initial condition of the system produces a relatively small and quantifiable or predictable change in an output state of the system. A nonlinear dynamical system, on the other hand, may exhibit a relatively sensitive dependence on system initial conditions. Thus, a relatively small or even a virtually unmeasurable difference in system initial conditions can result in nonpredictable system output states. Such output states may in some instances have relatively large differences between them despite the relatively small differences in initial conditions of the system.
This causes the output states of some dynamical systems to appear to be random in nature. Many activities which appear to be random in nature are actually examples of a deterministic phenomenon referred to as chaos. The phenomena that have been shown to exhibit chaos include but are not limited to the transition from fluid flow to turbulent flow in fluids, many types of mechanical vibrations, irregular oscillations, chemical reactions, a rise and fall of epidemics, the irregular dripping of a faucet, and the behavior of biological systems including human cardiac systems. Generally, chaotic systems are relatively sensitive to perturbations of their initial condition.
Typically, systems exhibiting a chaotic behavior are analyzed by developing a model of the system sufficiently detailed to identify one or more key parameters of the system. One problem with this approach, however, is that this technique is typically useful in systems for which a theoretical model is known and which do not display irreversible parametric changes. Such parametric changes, however, may sometimes themselves be the very changes causing the chaos.
It has been recognized that electrical signals produced by a human heart reflect the activity of a nonlinear dynamical system which may be described using chaos theory. Thus, the human heart may be referred to as nonlinear dynamical or chaotic system. Dynamical systems such as the heart can exhibit both periodic and chaotic behavior depending upon certain system parameters. These parameters appear as constants in mathematical equations describing the system. The chaotic behavior exhibited by the heart, however, is not immediately obvious when looking, for example, at an electrocardiograph (ECG) signal.
One way to observe the chaotic behavior of the heart has been to plot the interbeat spacing or its reciprocal (i.e. heart rate) at a time n against the interbeat spacing (or heart rate) at time n+1. Such a plot is referred to as a Poincare map or a return map. One problem with this technique, however, is that a relatively large amount of data is required to provide an accurate representation of the system. Furthermore, problems arise in collecting large amounts of data from biosystems. For example collection of a relatively large amount of bioelectric data on a human heart requires a human to wear a monitor or sensor for a relatively long period of time. Similarly, collection of a large number of human fluid samples requires a human to be struck with a syringe or other fluid drawing device. Moreover, relatively large processing power is required to analyze the large amount of data retrieved from the human subject or other bio system. The need to process such large amounts of data makes it relatively difficult to provide a real time processing system. Furthermore, a relatively large amount of storage capacity is required to store the large amount of collected data. Finally, the system under study may be non-stationary (i.e., may vary) over a long period of time and thus, a large data set collected over a long time interval may not accurately reflect the behavior of the system.
It would, therefore, be desirable to provide a technique which can be used to detect the presence of nonlinearity in a dynamical system. It would also be desirable to determine whether nonlinear dynamical systems are chaotic systems. It would further be desirable to provide a technique for diagnosing disease in biosystems by detecting the presence of chaos in a biosignal (e.g. based upon their nonlinear or chaotic behavior). For example, it would be desirable to diagnose heart disease or heart failure by detecting the presence of chaos in an ECG signal.